Second order elliptic equations with mixed boundary conditions

Abstract

We consider the mixed boundary value problem Au = f in Ω, B 0u = g 0 in Γ −, B 1u = g 1 in Γ + , where Ω is a bounded open subset of R n whose boundary Γ is divided into disjoint open subsets Γ + and Γ − by an ( n − 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on Ω and B j , for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on Γ + . The coefficients of the operators and Γ +, Γ − and ω are all assumed arbitrarily smooth. As announced in [ Bull. Amer. Math. Soc. 83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset T of the reals such that, if D s = {u ϵ H s(Ω): Au = 0} then for s ≢ = 1 2 ( mod 1), (B 0,B 1): D s → H s − 1 2 (Γ −) × H s − 3 2 (Γ +) is a Fredholm operator if and only if s ∈ T . Moreover, T = ⋂ xew T x, where the sets T x are determined algebraically by the coefficients of the operators at x. If n = 2, T x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, T x is either an open interval of length 1 or is empty; and finally, if n ⩾ 4, T x is an open interval of length 1.